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Absolutely divergent series and classes of Banach spaces

Published online by Cambridge University Press:  24 October 2008

William H. Ruckle
William H. Ruckle
Affiliation:
Department of Mathematical Sciences, Clemson University, Clemson, SC 29631, U.S.A.
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Extract

A normed space E is said to be series immersed in a Banach space X if for every absolutely divergent series nxn in E there is a continuous linear mapping T from E into X such that nTxn diverges absolutely. The theorem of Dvoretzky and Rogers(1) implies that a normed space E is series immersed in a finite dimensional space if and only if E itself is finite dimensional. In (4) and (9) it was shown that E is series immersed in lp 1 p < if and only if E is isomorphic to a subspace of Lp() for some measure . In particular, E is isomorphic to an inner product space if and only if it is series immersed in a Hilbert space. The property of series immersion was further studied in the papers (7) and (8). The main results in these two papers are conditions on X under which E series immersed in X would imply E locally immersed in X, a condition slightly stronger (formally) than E finitely representable in X.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 94 , Issue 2 , September 1983 , pp. 281 - 289
Copyright
Copyright © Cambridge Philosophical Society 1983

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References

REFERENCES

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